Green;s Second Identity fro Ructions That Differentiate Twice to a Multiple of the Function

Theorem
If
$\phi (x,y,z), \: \psi (x,y,z)$
be harmonic functions with continuous first and second partial derivatives on a region
$R$
containing a region
$D$
with surface
$S$
. If
$\phi (x,y,z), \: \psi (x,y,z)$
satisfy the equations
$\nabla^2 \phi = f \phi , \: \nabla^2 \psi = f \psi ,$
where
$f=f(x,y,z)$
in
$D$

Then
$\int \int_S \phi (\frac{\partial \psi}{\partial n} - \frac{\partial \psi}{\partial n} ) dS =0$

Proof
Green's Second Idenity can be written
$\int \int_S ( \phi \frac{\partial \psi}{\partial n} - \psi \frac{\partial \phi}{\partial n} ) dS = \int \int \int_D (\phi \nabla^2 \psi - \psi \nabla^2 \phi )dV$

Then
$\int \int_S ( \phi \frac{\partial \psi}{\partial n} - \psi \frac{\partial \phi}{\partial n} ) dS = \int \int \int_D (\phi \nabla^2 \psi - \psi \nabla^2 \phi )dV = \int \int \int_D (\phi f \psi - \psi f \phi ) dV =0$